Simple Cells Thurs. Jan. 25, 2018 1 Recall last lecture: DOG - - PowerPoint PPT Presentation

β–Ά
simple cells
SMART_READER_LITE
LIVE PREVIEW

Simple Cells Thurs. Jan. 25, 2018 1 Recall last lecture: DOG - - PowerPoint PPT Presentation

Apr 01, 2024 119 likes β€’544 views

COMP 546 Lecture 5 Orientation Selection 1: Simple Cells Thurs. Jan. 25, 2018 1 Recall last lecture: DOG (lateral inhibition), cross correlation image - - + - - - DOG 2 Example: an edge image ,

slide-1
SLIDE 1

1

COMP 546

Lecture 5

Orientation Selection 1: Simple Cells

  • Thurs. Jan. 25, 2018
slide-2
SLIDE 2

Recall last lecture:

DOG (lateral inhibition), β€œcross correlation”

2

+

  • -
  • image

DOG

slide-3
SLIDE 3

Example: an edge image

3

𝑦 𝐽 𝑦, 𝑧 𝐽 𝑦

slide-4
SLIDE 4

Example: an edge image

4

𝑦 𝐽 𝑦, 𝑧 +

  • -
  • +
  • -
  • +
  • -
  • +
  • -
  • +
  • -
  • 𝐸𝑃𝐻 ⨂ 𝐽 𝑦, 𝑧

+

  • -
  • DOG
slide-5
SLIDE 5

Mach Bands

Are they the result of lateral inhibition in the retina ?

5

slide-6
SLIDE 6

ASIDE: Mach bands are well known problem for interpreting x-ray

  • images. Very subtle changes in dark-bright must be detected and

the visual system is often fooled.

6

slide-7
SLIDE 7

Retinal ganglion cells encode image differences :

  • spectral (wavelength l) , β€œchromatic”
  • spatial (x,y)
  • temporal (t) --

will cover this next week

  • spectral-spatial (l, x, y) - Assignment 1
  • spectral-spatio-temporal (l, x, y, t) -
  • mit

7

slide-8
SLIDE 8

Assignment 1

8

R+ G- R- G+ Double opponent cells Single opponent cells R+ G-

slide-9
SLIDE 9

Early visual pathway: retina to cortex (V1)

Lateral geniculate nucleus (LGN)

9

slide-10
SLIDE 10

Left visual field Right visual field

10

(LGN)

slide-11
SLIDE 11

Polar coordinates on the retina

11

Vertical meridian Horizontal meridian

slide-12
SLIDE 12

Retinotopic Map

12

Definition: Cells in a visual area are spatially arranged in a retinotopic map if physically adjacent cells in that area have adjacent receptive fields (and hence encode image in adjacent regions of the retina) retina Some visual area in the brain e.g. LGN, V1

slide-13
SLIDE 13

LGN

V1 retina

13

Layers 1 and 2 (bigger receptive fields time dependent, no color opponency) Layers 3, 4, 5, 6 (small receptive fields, color

  • pponency)
slide-14
SLIDE 14

14

  • Polar coordinates in

primary visual cortex (V1)

right visual field

slide-15
SLIDE 15

15

functional magnetic resonance imaging (fMRI)

slide-16
SLIDE 16

The moving slide (see 35 sec and on...) http://www.youtube.com/watch?v=IOHayh06LJ4 3 minutes of exploration: https://www.youtube.com/watch?v=Cw5PKV9Rj3o

What do cells in V1 encode ?

(Hubel & Wiesel 1959)

16

slide-17
SLIDE 17

β€œSimple Cell”

17

Temporal effects to be discussed in lecture 7

slide-18
SLIDE 18

V1 Orientation Tuning Curve

18

slide-19
SLIDE 19

+

  • -
  • +
  • -
  • +
  • -
  • +
  • -
  • LGN

V1 retina

Hubel and Wiesel suggested this mechanism for elongated receptive field profile of V1 simple cell

19

+

  • -
  • +
  • -
  • R+ G-

R- G+

slide-20
SLIDE 20
  • + -
  • + -
  • + -
  • + -
  • +
  • +
  • +
  • +

β€œLine Detector” β€œEdge Detector” (even) (odd)

Model of orientation selectivity in V1

20

𝑀 =

𝑦,𝑧

𝑔 𝑦 βˆ’ 𝑦0 , 𝑧 βˆ’ 𝑧 𝐽 𝑦, 𝑧

Cell centered at 𝑦0, 𝑧0

slide-21
SLIDE 21

Cell response model:

half-wave rectification

21

Response (spike rate)

𝑀

Quasi-linear : cell response is linear over some range.

slide-22
SLIDE 22
  • + -
  • + -
  • + -
  • + -
  • +
  • +
  • +
  • +

+ - + + - + + - + + - + + - + - + - + -

Line Detector (even) Edge Detector (odd)

22

How to encode the negative values of ? (similar idea to last lecture)

𝑀

slide-23
SLIDE 23

β€œGabor” function: classical model of simple cell

23

Line (even) Edge (odd)

slide-24
SLIDE 24

1D Cosine Gabor

24

π»π‘π‘£π‘‘π‘‘π‘—π‘π‘œ π‘‘π‘π‘‘π‘—π‘œπ‘“

βˆ— =

π‘‘π‘π‘‘π‘—π‘œπ‘“ 𝐻𝑏𝑐𝑝𝑠

slide-25
SLIDE 25

1D Sine Gabor

25

βˆ— =

π‘‘π‘—π‘œπ‘“

π‘‘π‘—π‘œπ‘“ 𝐻𝑏𝑐𝑝𝑠

π»π‘π‘£π‘‘π‘‘π‘—π‘π‘œ

slide-26
SLIDE 26

(Sampled) Cosine

26

cos( 2𝜌 𝑂 𝑙𝑦 𝑦) 𝑓. 𝑕. 𝑙𝑦 = 8 𝑂 = 256

𝑙𝑦 is spatial frequency

slide-27
SLIDE 27

1D Cosine Gabor

27

1 2𝜌 𝜏 π‘“βˆ’ 𝑦2

2𝜏2

cos(2𝜌 𝑂 𝑙𝑦 𝑦)

βˆ— =

slide-28
SLIDE 28

1D Sine Gabor

28

1 2𝜌 𝜏 π‘“βˆ’ 𝑦2

2𝜏2

sin(2𝜌 𝑂 𝑙𝑦 𝑦)

βˆ— =

slide-29
SLIDE 29

2D cosine

29

cos 2𝜌 𝑂 (𝑙𝑦 𝑦 + 𝑙𝑧 𝑧) 𝑓. 𝑕. 𝑙𝑦 = 4 𝑙𝑧 = 0 𝑂 = 256

slide-30
SLIDE 30

2D sine

30

sin 2𝜌 𝑂 (𝑙𝑦 𝑦 + 𝑙𝑧 𝑧) 𝑓. 𝑕. 𝑙𝑦 = 8 𝑙𝑧 = 2 𝑂 = 256

slide-31
SLIDE 31

model of simple cell: 2D Gabor

31

𝐻 𝑦, 𝑧, 𝜏

cos 2𝜌 𝑂 (𝑙𝑦 𝑦 + 𝑙𝑧 𝑧)

𝐻 𝑦, 𝑧, 𝜏

sin 2𝜌 𝑂 (𝑙𝑦 𝑦 + 𝑙𝑧 𝑧)

𝑓. 𝑕. 𝑙𝑧 = 0 𝑓. 𝑕. 𝑙𝑧 = 0

slide-32
SLIDE 32

32

slide-33
SLIDE 33

33

  • What is the response of a family of Gabor cells to a

single image ?

e.g. Consider shifted versions of the Gabor cell.

  • What is the response of a single Gabor cell to a

parameterized family of images ?

e.g. thin white line at different positions in receptive field

slide-34
SLIDE 34

34

What is the response of a family of Gabor cells to a single image?

slide-35
SLIDE 35

35

cross correlation with (four) cosine Gabors

slide-36
SLIDE 36

36

cross correlation with (four) sine Gabors

slide-37
SLIDE 37

37

  • What is the response of a family of Gabor cells to a

single image ?

e.g. Consider shifted versions of the Gabor cell.

  • What is the response of a single Gabor cell to a

parameterized family of images ?

e.g. thin white line at different positions in receptive field

slide-38
SLIDE 38

38

𝑀 ≑

𝑦 ,𝑧

𝑑𝑝𝑑𝐻𝑏𝑐𝑝𝑠 𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦π‘₯β„Žπ‘—π‘’π‘“ π‘šπ‘—π‘œπ‘“

slide-39
SLIDE 39

39

𝐻 𝑦, 𝑧, 𝜏 cos 2𝜌 𝑂 (𝑙𝑦 𝑦)

Non-zero only at 𝑦 position

  • f vertical line

𝑀 ≑

𝑦 ,𝑧

𝑑𝑝𝑑𝐻𝑏𝑐𝑝𝑠 𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦π‘₯β„Žπ‘—π‘’π‘“ π‘šπ‘—π‘œπ‘“

slide-40
SLIDE 40

40

𝐻 𝑦, 𝑧, 𝜏 π‘‘π‘—π‘œ 2𝜌 𝑂 (𝑙𝑦 𝑦)

Non-zero only at 𝑦 position

  • f vertical line

𝑀 ≑

𝑦 ,𝑧

π‘‘π‘—π‘œπ»π‘π‘π‘π‘  𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦π‘₯β„Žπ‘—π‘’π‘“ π‘šπ‘—π‘œπ‘“

slide-41
SLIDE 41

41

Gaussian envelope (discuss next lecture)